The Electromagnetic Rail Launcher · AI Generator #GEN-5C2A13D6

Physics · electromagnetism

Problem

A military research facility is testing an electromagnetic rail launcher that accelerates projectiles using magnetic forces. The launcher consists of two parallel conducting rails separated by distance

d = 0.50\;\text{m}

and connected to a capacitor bank. A conducting projectile of mass

m = 2.0\;\text{kg}

and resistance

R = 8.0\;\text{m}\Omega

slides along the rails, completing the circuit. The capacitor bank has capacitance

C = 0.40\;\text{F}

and is initially charged to voltage

V_0 = 2000\;\text{V}

. The rails and connecting wires have negligible resistance and inductance. As the projectile accelerates down the rails, it cuts through magnetic field lines, inducing a back-EMF that opposes the driving current. The magnetic field

B

at the location of the projectile is proportional to the instantaneous current:

B = \beta I

, where

\beta = 0.050\;\text{T/A}

. This relationship accounts for the field produced by the current-carrying rails themselves. (a) Find the maximum velocity

v_{\text{max}}

that the projectile can theoretically achieve (when acceleration becomes zero). (b) Calculate the kinetic energy of the projectile when it reaches

90\%

of its maximum velocity.

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